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Staff Working Paper/Document de travail du personnel 2016-5

Macroeconomic Uncertainty Through the Lens of Professional Forecasters

by Soojin Jo and Rodrigo Sekkel

2

Bank of Canada Staff Working Paper 2016-5

February 2016

Macroeconomic Uncertainty Through the Lens of Professional Forecasters

by

Soojin Jo1 and Rodrigo Sekkel2

1Canadian Economic Analysis Department Bank of Canada

Ottawa, Ontario, Canada K1A 0G9 sjo@bankofcanada.ca

2Financial Markets Department

Bank of Canada Ottawa, Ontario, Canada K1A 0G9

rsekkel@bankofcanada.ca

ISSN 1701-9397 © 2016 Bank of Canada

ii

Acknowledgements

We thank Natsuki Arai, Greg Bauer, Christiane Baumeister, Marcelle Chauvet, Peter Christoffersen, Rafaella Giacomini, Lutz Kilian, James D. Hamilton, Monica Jain, Alejandro Justiniano, Kajal Lahiri, Michael McCracken, Ulrich Mueller, Xuguang Sheng, Gregor Smith, and Alexander Ueberfeldt for useful comments and suggestions, as well as seminar participants at American University, the Bank of Canada, U.S. Bureau of Economic Analysis, Hamilton College, SUNY Albany, University of Alberta, the 2014 Workshop on Uncertainty and Economic Forecasting (UCL), 2014 Canadian Economic Association, 2014 Computing in Economics and Finance, 2014 IAAE meetings, and the 2014 Society for Economic Measurement meeting, the 2015 CIRANO Real-Time Workshop, and the St. Louis Fed Applied Time Series Econometrics Workshop.

iii

Abstract

We analyze the evolution of macroeconomic uncertainty in the United States, based on the forecast errors of consensus survey forecasts of different economic indicators. Comprehensive information contained in the survey forecasts enables us to capture a real-time subjective measure of uncertainty in a simple framework. We jointly model and estimate macroeconomic (common) and indicator-specific uncertainties of four indicators, using a factor stochastic volatility model. Our macroeconomic uncertainty has three major spikes, aligned with the 1973–75, 1980, and 2007–09 recessions, while other recessions were characterized by increases in indicator-specific uncertainties. We also demonstrate for the first time in the literature that the selection of data vintages substantially affects the relative size of jumps in estimated uncertainty series. Finally, our macroeconomic uncertainty has a persistent negative impact on real economic activity, rather than producing “wait-and-see” dynamics.

JEL classification: C38, E17, E32 Bank classification: Business fluctuations and cycles; Econometric and statistical methods

Résumé

Nous analysons l’évolution de l’incertitude macroéconomique aux États-Unis, à partir des erreurs de projection de différents indicateurs économiques. Celles-ci donnent des renseignements détaillés qui nous permettent de dégager une mesure subjective en temps réel de l’incertitude dans un cadre de référence simple. Nous avons à la fois modélisé et évalué l’incertitude macroéconomique (commune) et l’incertitude propre à quatre indicateurs, à l’aide d’un modèle factoriel à volatilité stochastique. D’après notre analyse, l’incertitude macroéconomique connaît trois pics principaux, qui coïncident avec les récessions de 1973-1975, de 1980 et de 2007-2009. Les autres récessions se caractérisent par une hausse de l’incertitude inhérente à chacun des indicateurs. Pour la première fois dans la littérature, nous démontrons également que le choix de la cuvée de données influe considérablement sur l’importance relative des sauts dans la série d’incertitudes évaluées. En dernier lieu, loin d’instaurer une dynamique attentiste, l’incertitude macroéconomique observée a une incidence défavorable persistante sur l’activité économique réelle.

Classification JEL : C38, E17, E32 Classification de la Banque : Cycles et fluctuations économiques; Méthodes économétriques et statistiques

iv

Non-Technical Summary

This paper analyzes the evolution of macroeconomic uncertainty in the United States by using consensus survey forecasts to estimate a subjective measure of uncertainty. Survey forecasts are not tied to any particular econometric model and have often been found to be more accurate than those forecasts based on the econometric models, especially for short horizons. We apply a factor stochastic volatility model to forecast errors of a set of economic indicators from the Survey of Professional Forecasters conducted by the Federal Reserve Bank of Philadelphia. Our parsimonious framework jointly models the common (macroeconomic) and indicator-specific uncertainties in one step.

Our estimated macroeconomic uncertainty is significantly more persistent than traditional proxies of uncertainty, with all major spikes associated with deep economic recessions (i.e., the 1973–75, 1980 and 2007–09 recessions). However, other recessions (i.e., the 1990–91 and 2001 recessions) are still notable in the dynamics of some of the idiosyncratic uncertainties.

In addition, a vector autoregression (VAR) analysis shows that positive innovations to macroeconomic uncertainty are associated with decreases in a variety of real economic variables; investment, non-durable and durable consumption, and GDP all decline after an increase in macroeconomic uncertainty. The impact of uncertainty shocks is not only sizable but also highly persistent. We do not find any evidence supporting subsequent overshooting in economic activity as the uncertainty shock dissipates. This contrasts with the findings from previous VAR analyses using traditional proxies for macroeconomic uncertainty, such as the VIX.

We also examine the impact of data revisions on the estimation of economic uncertainty. We find large differences in the relative size of jumps in uncertainty series depending on the specific vintage of the data. Hence, a data vintage selection can be very important, especially when the interest lies in estimating the relative level of macroeconomic uncertainty over time. To the best of our knowledge, our paper is the first study to document the importance of data revision to the estimation of uncertainty. Together, our findings deepen our understanding of the evolution of macroeconomic uncertainty and its relation to the macroeconomy.

1 Introduction

The literature on the impacts of uncertainty on real economic activity has recently witnessed

a rapid growth following the Great Recession.1 Many studies have aimed at empirically

quantifying the effect of uncertainty. Central to these studies is a need for a measure

of time-varying uncertainty, since uncertainty is not directly observable. Accordingly, a

number of proxies for uncertainty have been used. Bloom (2009) has pioneered the use

of the VIX, the implied stock market volatility based on the S&P index. Alternatively,

Bloom et al. (2012) use the cross-sectional dispersion of total factor productivity shocks.

Another popular proxy is the cross-sectional disagreement of individual forecasts, as in

Bachmann et al. (2013). A proxy constructed using word searches from newspaper articles

is proposed in Alexopoulos and Cohen (2009). Finally, the Baker et al. (2013) Economic

Policy Uncertainty index combines news article counts with the number of federal tax code

provisions set to expire as well as the forecast disagreement.

This paper analyzes the evolution of macroeconomic uncertainty in the United States by

using consensus survey forecasts for the estimation of a subjective measure of uncertainty.

In particular, we use the median forecasts of four economic indicators from the Survey of

Professional Forecasters (SPF) conducted by the Federal Reserve Bank of Philadelphia.

Using survey forecasts provides the following advantages. First, they are not tied to any

particular econometric models. This implies that the selection and estimation of a spe-

cific forecasting model to obtain forecast errors are not necessary, and hence the overall

uncertainty estimation procedure is simplified to a great extent. Second, subjective fore-

casts have often been found to be more accurate than forecasts from econometric models.2

Professional forecasters base their forecasts on various econometric models, leading indi-

cators and surveys, as well as a large number of available macroeconomic and financial

data, encompassing information from a wide range of sources (see Zarnowitz and Braun

1993). Therefore, survey forecasts provide an effective way of removing expected variation

1See, for example, Bloom (2009), Arellano et al. (2012), Caggiano et al. (2014), Aastveit et al. (2013),and Mumtaz and Surico (2013), among many others.

2For example, Ang et al. (2007) and Faust and Wright (2013) document the advantage of surveys overforecasting models for inflation. Aiolfi et al. (2011) study the optimal combination of the two types offorecasts for different indicators, and find that combinations always improve over time-series models, butstill fail to systematically improve on the survey forecasts alone.

2

in macroeconomic series. As highlighted by Jurado et al. (2015), it is crucial to remove the

predictable component of macro series when estimating macroeconomic uncertainty so as

not to attribute some of the predictable variability to unpredictable shocks.

Based on various economic indicators from the SPF, we define macroeconomic uncer-

tainty as the conditional time-varying standard deviation of a factor that is common to

forecast errors for four macroeconomic indicators such as gross domestic product, unem-

ployment, industrial production and housing starts. In other words, an increase in macroe-

conomic uncertainty implies a higher probability that such economic variables simultane-

ously deviate from their conditional survey forecasts. This idea is effectively captured by a

factor stochastic volatility (FSV) model, first developed by Pitt and Shephard (1999). The

stochastic volatility process has been widely adopted in finance literature.3 More recently, it

has been also often employed in macroeconomic analysis to model the time-varying volatil-

ity of macroeconomic variables.4 In our application, we use stochastic volatility processes

in a factor model structure to provide both time-varying common (macroeconomic) and

indicator-specific uncertainty indexes. The indexes are jointly modeled and consistently

estimated in one step.

Our estimated macroeconomic uncertainty is significantly more persistent than tradi-

tional proxies of uncertainty (e.g., the VIX). In particular, all major spikes of uncertainty

are associated with episodes of economic recessions, i.e., the 1973–75, 1980, and 2007–09

recessions, similar to the findings in Jurado et al. (2015). However, other recessions (i.e.,

the 1990–91 and 2001 recessions) are still notable in the dynamics of some of the idiosyn-

cratic uncertainties, but were not picked up by the macroeconomic uncertainty series. This

suggests that increases in uncertainty during these periods were not as broad-based as

during the 1973–75, 1980, and 2007–09 recessions.

In addition, vector autoregressions (VAR) show that innovations to macroeconomic

uncertainty generate negative effects on a variety of real economic variables; investment,

non-durable and durable consumptions, and GDP all decline after an increase in macroe-

conomic uncertainty. The impact of uncertainty shocks is not only sizable but also highly

3See, e.g., Collin-Dufresne and Goldstein (2002), Heston (1993), and Kim et al. (1998).4See Caldara et al. (2012), Jo (2014), Mumtaz and Zanetti (2013), Primiceri (2005), and Justiniano and

Primiceri (2008), among others.

3

persistent, in line with the findings in Jurado et al. (2015) and, to a lesser magnitude, those

in Bachmann et al. (2013). In addition, we do not find any evidence supporting subsequent

overshooting in economic activity as the uncertainty shock dissipates. This contrasts with

previous VAR analyses using traditional proxies for macroeconomic uncertainty, such as the

VIX in Bloom (2009) and Caggiano et al. (2014), where the negative effects of uncertainty

dissipate quickly, followed by strong rebounds.

We also examine the impact of data revisions on the estimation of economic uncertainty.

To the best of our knowledge, ours is the first paper so to do. As macroeconomic variables

are constantly revised, uncertainty measures based on the most recent data vintage use a

different information set than was previously available to professional forecasters. We find

large quantitative differences in the uncertainty series with real-time and revised forecast

errors. More specifically, the 1973–75 and 1980 recessions exhibit the largest jumps in our

baseline uncertainty index with forecast errors based on the initial release. Yet, a data

revision of four underlying series conducted within one quarter after the initial release

pushes the uncertainty jump accompanying the 1980 recession up further, leaving its peak

at the highest level throughout the entire sample period. On the contrary, uncertainty

is at its highest in the Great Recession when the final data vintage is used, although

overall dynamics remain similar. Hence, the use of a specific data vintage can be very

important, especially when the interest lies in estimating the relative level of macroeconomic

uncertainty over time.

Our paper shares some similarities with recent studies focusing on estimating macroe-

conomic uncertainty. For example, Jurado et al. (2015) fit a factor model to a large cross-

section of macroeconomic and financial variables to generate forecasts. They assume that

the volatilities of individual forecast errors follow a univariate stochastic volatility process,

whose average becomes macroeconomic uncertainty. Rossi and Sekhposyan (2015) use fore-

cast surveys and econometric models for GDP and inflation to construct an uncertainty

index from the unconditional historical distribution of forecast errors. Scotti (2013) exploits

survey forecasts and creates an uncertainty index as the weighted sum of the squared fore-

cast errors for different indicators. Our approach complements this literature by combining

survey forecasts of different indicators in a factor model with stochastic volatility. Thus, we

4

jointly estimate indicator-specific as well as macroeconomic uncertainties, providing clear

insights about co-movements among indicators. As a result, we are able to shed light on

the time-varying contribution of both aggregate and indicator-specific uncertainty for the

total variations of the forecasting errors of each indicator.

Our paper is also related to the recent development of literature that applies FSV mod-

els to financial and macroeconomic analysis. The FSV model has been widely used in the

finance literature as it can parsimoniously capture the variance and covariance of various

financial time series using a low-dimensional common factor (Kastner et al. 2014).5 On the

macroeconomic side, Carriero et al. (2015) introduce a VAR model where macroeconomic

variables share a common stochastic volatility factor. Unlike our framework, idiosyncratic

volatility is not modeled, since the variations of individual volatilities over time are com-

pletely determined by the common factor. Mumtaz and Theodoridis (2015) estimate a

dynamic factor model with stochastic volatility with a panel of OECD countries to de-

compose the time-varying volatilities of macro and financial variables to internationally

common, country-specific and variable-specific uncertainties. Finally, Del Negro and Otrok

(2008) develop a dynamic factor model with time-varying factor loadings and stochastic

volatility to study international business cycles. One common feature in the previous stud-

ies is that the evolution of the conditional means of the variables is parametrically linked

to the factors. In contrast, our approach fits the FSV model directly to survey forecast

errors without jointly modeling the conditional mean along with conditional volatilities,

for the first time in the literature to the best of our knowledge. Instead, we outsource

the modeling of the conditional mean to professional forecasters. Hence, we provide an

estimate of common and idiosyncratic uncertainties through the perspective of professional

forecasters.

The rest of the paper is organized as follows. Section 2 introduces the data set. Section

3 provides an exposition of our econometric model. The next section presents results. A

VAR analysis in Section 5 shows the responses of real economic activity to innovations in

our uncertainty measure. Finally, Section 6 concludes.

5See, e.g., Aguilar and West (2000) and Chib et al. (2009).

5

2 Data

We use the data from the U.S. SPF, the oldest quarterly survey of macroeconomic fore-

casts in the U.S. The survey was initially introduced by the National Bureau of Economic

Research and the American Statistical Association in 1968, which was then taken over

by the Federal Reserve Bank of Philadelphia in June 1990. Surveyed panelists include

forecasters from large corporations, Wall Street financial firms, economic consulting firms,

and university research centers. We use four variables that have been part of the survey

since its inception: real GDP (GDP), unemployment rate (UR), industrial production (IP)

and housing starts (HS). With the selected variables, we can obtain a long time series of

macroeconomic uncertainty since 1968Q4 until 2015Q2.

Since forecasters are surveyed on a quarterly basis, the most recent quarter of data in

their information set is the previous quarter. The forecast submission deadline of the survey

tends to occur close to the middle of the quarter (after the Bureau of Economic Analysis

advance report of the national income and product accounts [NIPA], which contains the first

estimates of the previous quarter’s GDP). Hence, for macroeconomic variables released on

a monthly basis, forecasters have access to the first month’s realized data for the current

forecast horizon before the survey is submitted. We use their one-step-ahead forecasts,

namely their nowcasts, in order to construct the forecast errors.

The calculation of forecast errors at any point in time is contingent on the realized

value of the series. The NIPA data go through substantial revisions, and these revisions

can ultimately affect our measurement of uncertainty. Thus, we use the first release, revised

data available in one and five quarters after the initial release, as well as the final 2015Q2

vintage data to compute three possible values of forecast errors. When calculating the

forecasting errors, we use the consensus forecasts to minimize potential influences from

individual forecasting biases.6

6Arai (2014) finds that the SPF consensus forecasts for GDP growth present no systematic biases.

6

3 Factor Stochastic Volatility Model

We use the FSV model of Pitt and Shephard (1999) to estimate macroeconomic as well as

idiosyncratic uncertainty indexes. First, we define the forecasting error of a variable i in

period t, denoted as εi,t, as follows:

εi,t = xi,t −median{E1[xi,t|It], · · · , EJ [xi,t|It]}, (1)

where xi,t is the realization of variable i in time t, Ej[xi,t|It] is forecaster j’s expectation of

variable i for quarter t, and median{E1[xi,t|It], · · · , EJ [xi,t|It]} captures the median forecast

across total J forecaster in each quarter.7 Again, one of the key differences of our measure

from other uncertainty indexes based on a particular forecasting model is that we obtain

E[xi,t|It] from the consensus forecasts instead of using forecasts from a specific econometric

model. The information set (It) also has the same time-subscript t, as it contains infor-

mation obtained until the middle of the quarter t. That is, for indicators of quarterly

frequency, It includes the first NIPA estimate of xi,t−1 that is available only in the middle

of the quarter t. However, as discussed in the previous section, for monthly macroeconomic

indicators such as IP, the first month’s value in the quarter t is included in It along with

the first NIPA release of xi,t−1.

Next, we postulate that the forecasting error of a macroeconomic series i has a factor

structure:

εt = λft + ut, (2)

where εi,t = [ε1,t, · · · , εn,t]′ is a (n × 1) vector of forecasting errors; λ = [λ1, · · · , λn]′ is a

vector of factor loadings; ft is a common factor across different i’s; and ut = [u1,t, · · · , un,t]′

is a vector of idiosyncratic errors, capturing indicator-specific variations. Equation (2)

implies that there is a factor that drives the common dynamics across the forecasting

errors of total n economic indicators, and that ut captures the remaining indicator-specific

variations.

7We use the median and not the mean forecast across survey respondents to minimize the effect ofoutliers, especially in the earlier part of the sample.

7

As in a standard FSV model (e.g., Pitt and Shephard 1999 and Chib et al. 2006), we

further assume that ut and ft are conditionally independent Gaussian random vectors. ut

ft

|Σt, hf,t ∼ N

0,

Σt 0

0 hf,t

, (3)

where Σt are a n × n diagonal matrix of time-varying idiosyncratic volatilities. In other

words,

Σn,t =

h1,t 0 · · · 0

0 h2,t · · · 0...

.... . .

...

0 0 · · · hn,t

.

The common as well as indicator-specific volatilities follow independent stochastic volatility

processes:

log hf,t = log hf,t−1 + σfηf,t

log hi,t = log hi,t−1 + σiηi,t, (4)

where σf and σi’s are time-invariant parameters determining the variability of the volatil-

ities, and ηf,t and ηi,t capture innovations to volatility. One reason that the stochastic

volatility model has been widely adopted in literature is its flexibility due to the innovation

term in the volatility process in addition to that of the first moment. Applied to our set-up,

it is possible to examine shocks to the volatility process, which can be uncorrelated with

unexpected variations in the level of forecast errors.8

The key estimate of interest is the time-varying standard deviations of the factor, i.e.,

{√hf,t }, which we define as a measure of the macroeconomic uncertainty: the time series of

macroeconomic uncertainty captures the volatility of a common driver that simultaneously

affects the magnitude of forecasting errors across different real activity indicators. Other

8Hence, the resulting shock processes can be included in a univariate regression framework to gauge theeffects of uncertainty shocks in a straightforward way. For example, the shock processes can be used toobtain impulse responses to an uncertainty shock by local projections a la Jorda (2005).

8

important estimates are the time-varying standard deviations of idiosyncratic errors, i.e.,

{√hi,t }. These idiosyncratic volatility series will capture the size of indicator-specific

shocks, which are orthogonal to the common factor by construction. It is worthwhile to

note again that our framework hence yields both common and indicator-specific uncertainty

indexes, which are consistently modeled and estimated in one step.

The model is estimated using Bayesian methods, since it features high dimensionality as

well as non-linearity. The Bayesian methods deal with such features by separating param-

eters into several blocks, which greatly simplifies the estimation process. In particular, the

Markov Chain Monte Carlo (MCMC) algorithm breaks the parameters into several blocks

and repeatedly draws from their conditional posterior distributions in order to simulate the

joint posterior distribution. While details regarding the MCMC algorithm is provided in

Appendix A, here we briefly summarize steps in the estimation procedure as follows:

1. Assign initial values for λ, {fT}, σf , {hf}, and σi and {hi} for all i.

2. Draw λ from p(λ|εT , fT , σf , σi, hTf , hTi ).

3. Draw {fT} from p(fT |εT , λ, σf , σi, hTf , hTi ).

4. Draw σf and σis from p(σ|εT , fT , hTf , hTi ).

5. Draw {hf}, and {hi}’s from p(h|εT , fT , σf , σi).

6. Go to step 2.

The conditional independence across t and i as well as of ft and ui,t’s simplifies the

estimation procedure. More specifically, steps in the Gibbs sampler, such as drawing factor

loadings, volatility states and the variance of volatilities (i.e., steps 2, 4 and 5) break down

to drawing from a univariate process. Likewise, step 3 becomes estimating a Bayesian

regression of forecast errors on factor loadings in every period t with known error variance.

We collect 5, 000 draws by storing every 10th draw in order to avoid potential auto-

correlation across draws. Thus, we iterate over the sampler a total of 80, 000 times, but

discard the first 30, 000 draws of parameters. We follow the common identification scheme

of a factor model that sets the first factor loading (of GDP) equal to unity. However, the

resulting baseline index of macroeconomic uncertainty is robust to equalizing other factor

loadings to one or to changing the ordering of variables. The choice of prior distributions

9

Table 1: Summary Statistics of Posterior Draws of Factor Loadings

GDP IP UR HS

Median 1 1.39 −1.07 0.40Std. Dev. - 0.33 0.25 0.16

Note: This table shows the median and standard deviations calculated from the posterior draws of fourfactor loadings. Since our identification strategy is to set the loading of GDP to unity, the standard devi-ation is not reported for GDP.

and their parameter values is very similar to that of Pitt and Shephard (1999). A de-

tailed description of the prior distribution setup and the MCMC algorithm is provided in

Appendix A.

4 Results

4.1 Estimated Macroeconomic and Idiosyncratic Uncertainties

We plot our baseline macroeconomic uncertainty series in Figure 1: the solid line is the

median posterior draw ({√hf,t }Tt=1) and the shaded area represents the 95 percent posterior

credible set. For our baseline estimates, we use the first data release to calculate the

forecasting errors.9

There are three main spikes in macroeconomic uncertainty, all associated with deep

recessions. The first spike was observed during the 1973–75 recession, the second during

the 1980 recession, and the last one during the recent Great Recession. The highest increase

in macroeconomic uncertainty occurred during the 1980 recession. It is also clear from the

figure that, in general, the level of macroeconomic uncertainty was significantly higher in

the 1968–85 period than from 1985 until the Great Recession, consistent with the findings

in Kim and Nelson (1999) and McConnell and Perez-Quiros (2000). The index shows some

increase around the 1991 recession, but it is a small one in comparison with the three critical

spikes. The 2001 recession, on the other hand, was accompanied by very mild increases in

macroeconomic uncertainty.

Table 1 reports the median posterior draws of factor loadings. For identification of a

9We assess the effect of data revisions on our uncertainty measure in Section 4.3.

10

Figure 1: Estimated Macroeconomic Uncertainty Series

1970 1975 1980 1985 1990 1995 2000 2005 2010 20150

0.5

1

1.5

2

2.5

Note: This figure plots the baseline macroeconomic uncertainty series estimated using the first-releaseddata. The black solid line is the median posterior draws of the time-varying standard deviation of acommon factor across forecasting errors of four macroeconomic indicators. The shaded area is the 95percent posterior credible set.

11

factor and factor loadings, we set the factor loading of GDP to unity, as mentioned in the

previous section. However, the relative sizes of the factor loadings and subsequently the

estimated series of uncertainty are robust to different normalization.10 We find that housing

starts load least on the common factor, while the loadings of the other three variables are

in a comparable range.

Using the median posterior draws, we further examine how much of the total variation

in the forecasting errors of each indicator is driven by the common versus idiosyncratic

volatilities. This is calculated by using the factor structure of our model. In particular,

our model implies a total variance of each variable in each period, var(εi,t), to be

var(εi,t) = var(λift + ui,t)

= λ2i var(ft) + var(ui,t)

= λ2ihf,t + hi,t, (5)

as the factor and idiosyncratic error terms are assumed to be uncorrelated. We then

measure the size of the total common variation driven by macroeconomic uncertainty in

each period as λ2i var(ft) = λ2ihf,t, incorporating the heterogeneity due to the difference in

factor loadings. Next, we compare√var(εi,t) and λi

√hf,t to investigate the contributions

of the common and idiosyncratic uncertainties.

Figure 2 plots the estimated total variation with the part explained by macroeconomic

uncertainty for each indicator. For most variables except housing starts, the most notable

spikes in the total variation are driven to a large extent by macroeconomic uncertainty.

Moreover, recessions that were not accompanied by distinct increases in macroeconomic

uncertainty, as in the 1991 and 2001 recessions, do show up in the total variations of

unemployment and, to a smaller extent, GDP and industrial production.

During the period of 1985 to 2007, when the baseline macroeconomic uncertainty index

was relatively subdued, idiosyncratic volatilities explain a large fraction of the total volatil-

ities of the different series. In the case of GDP, the share of idiosyncratic volatility takes

10In addition, while the medians change depending on which vintage is used to calculate forecast errors,the relative sizes of the most factor loadings also remain robust to the change of the data vintages. Theresults based on different data vintages are available upon request.

12

Figure 2: Total Variation versus Macro Variation

1970 1975 1980 1985 1990 1995 2000 2005 2010 20150

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Total VariationMacro Variation

1970 1975 1980 1985 1990 1995 2000 2005 2010 20150

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Total VariationMacro Variation

(a) GDP (b) IP

1970 1975 1980 1985 1990 1995 2000 2005 2010 20150

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Total VariationMacro Variation

1970 1975 1980 1985 1990 1995 2000 2005 2010 20150

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Total VariationMacro Variation

(c) Unemployment (d) Housing Starts

Note: This figure shows how much of total variation of each variable is explained by the macroeconomicuncertainty. The grey line is the total variation of one variable (defined as standard deviation), and theblack line is the macroeconomic uncertainty multiplied by a factor loading. All calculations are based onthe median posterior draws.

13

more than 50 percent of the total variation of GDP forecasting errors during this period.

Finally, it is interesting to note that the variation of housing starts contributes the least

to the macroeconomic uncertainty, and it leads the total volatility of the other indicators.

Interestingly, the idiosyncratic uncertainty of housing starts was on an increasing trend

from the mid-1990s to the Great Recession.

4.2 Comparison with Other Common Measures of Uncertainty

Various attempts have been made recently to quantify economic uncertainty: we compare

our baseline measure with a measure based on forecast disagreements, the VIX, and finally,

the one introduced in Jurado et al. (2015).

Forecast disagreement

A widely used proxy for uncertainty based on survey forecasts is forecast disagreement,

commonly measured as the interquartile range (e.g., Bachmann et al. 2013). Underlying

this practice is the assumption that predictions of forecasters are more likely to be close

to each other when economic uncertainty is low. However, forecast disagreement may just

reflect heterogeneous, but not uncertain, beliefs.11 Figure 3 plots the forecast disagreements

of the four economic series underlying our baseline model, along with the total variations

of median forecast errors calculated as in (5). The forecast disagreement is measured as

the interquartile ranges, i.e., the 75th percentile minus the 25th percentile of individual

forecasts.

In general, for each indicator, both disagreement and total variation from our model

exhibit similar dynamics over time. However, disagreements are notably more volatile than

total variations. In addition, all variables show episodes where the disagreement and total

variation deviate substantially from each other.

We further investigate the relationship between our macroeconomic uncertainty measure

and an analogous, disagreement-based proxy of macroeconomic uncertainty. We take the

averages of disagreement of the four indicators to construct a comparable series. Panel

11See, for example, Mankiw et al. (2004), Lahiri and Sheng (2010), and Sill (2012) for more detaileddiscussions of measuring uncertainty using forecast disagreement.

14

Figure 3: Stochastic Volatility and Dispersions

1970 1975 1980 1985 1990 1995 2000 2005 2010−2

−1

0

1

2

3

4

5Total VolatilityDisagreement

1970 1975 1980 1985 1990 1995 2000 2005 2010−2

−1

0

1

2

3

4

5Total VolatilityDisagreement

(a) GDP (b) IP

1970 1975 1980 1985 1990 1995 2000 2005 2010−2

−1

0

1

2

3

4

5Total VolatilityDisagreement

1970 1975 1980 1985 1990 1995 2000 2005 2010−2

−1

0

1

2

3

4

5Total VolatilityDisagreement

(c) Unemployment (d) Housing Starts

Note: This figure compares forecast disagreement from the SPF to total variation of underlying individualseries calculated as (5). The forecast disagreement is measured as the interquartile ranges, i.e., the 75thpercentile minus the 25th percentile of individual forecasts. All series are standardized.

15

(a) in Figure 4 presents the resulting average disagreement series along with our baseline

measure of macroeconomic uncertainty. The average disagreement series evolves closer to

our macroeconomic index, compared with the dynamics of individual disagreements; the

three major spikes in disagreement coincide with the three main episodes of uncertainty

increases in our baseline index.

VIX

Since employed in Bloom (2009), the VIX index has been one of the most commonly used

measures of uncertainty. The VIX index quantifies implied volatility using a weighted

average of the S&P 500 call-and-put option prices for the next 30 days and is available on a

daily basis. Thus, it mainly captures the financial volatility. Panel (b) in Figure 4 plots the

VIX along with our baseline measure of macroeconomic uncertainty after standardization

for comparison. The VIX is more volatile and has more peaks, some of which are specifically

due to financial market episodes that did not spill over to the real side of economy. More

importantly, the VIX collectively reflects both stock market uncertainty and risk aversion,

and, as a result, some of the jumps are mainly due to increases in risk aversion rather than

those in uncertainty, as shown in Bekaert et al. (2013).

Jurado et al. (2015)’s macroeconomic uncertainty index

The macroeconomic uncertainty measure constructed in Jurado et al. (2015) is based on

the same definition of uncertainty: it is the time-varying conditional second moment of

unexpected variations that affects forecasting errors of a number of economic indicators

simultaneously. Our approach and Jurado et al.’s measure share an additional similarity,

as both employ a stochastic volatility model as well as a factor structure. However, Jurado

et al. use the factor structure mainly for forecast generation: a large (dynamic) factor model

is built encompassing 132 macroeconomic and 147 financial series, where the forecasts for

all series are computed. These forecasts are used to calculate forecast errors of each series,

after which a univariate stochastic volatility process is fitted to 132 macroeconomic series.

Finally, the estimated volatilities are averaged to form a macroeconomic uncertainty index.

Thus, our approach contrasts with that of Jurado et al. (2015) in that, in our approach,

16

Figure 4: Comparison with Other Measures

1970 1975 1980 1985 1990 1995 2000 2005 2010 2015

−1

0

1

2

3

4

5

6 BaselineDispersion

1970 1975 1980 1985 1990 1995 2000 2005 2010 2015

−1

0

1

2

3

4

5

6 BaselineVIX

(a) Disagreement (b) VIX

1970 1975 1980 1985 1990 1995 2000 2005 2010 2015

−1

0

1

2

3

4

5

6 BaselineJNL

(c) JNL

Note: This figure compares our baseline macroeconomic series to other widely used indexes of uncertainty.Panel (a) plots the averages of forecast disagreement of the four economic indicators; (b) the VIX; and (c)the Jurado et al. (2015) uncertainty series, along with our baseline measure, respectively. All series arestandardized.

17

(i) forecasts are simply taken from a survey and (ii) macroeconomic and indicator-specific

uncertainties are jointly modeled and estimated in one step. In addition, the number

of underlying economic indicators is drastically different: while our measure is based on

four indicators, Jurado et al. take into account close to 300. Despite the differences, it

is surprising that both measure show very similar dynamics and identify the same three

main episodes of high uncertainty, as illustrated in Panel (c) in Figure 4. This reflects the

extensive information content in survey forecasts: Zarnowitz and Braun (1993) document

that professional forecasters in the SPF generate forecasts based on various econometric

models, leading indicators and surveys as well as a number of available macroeconomic

and financial data, integrating information from a wide range of sources. Consequently,

our approach removes predictable variations very effectively, providing a parsimonious yet

efficient tool for the estimation of uncertainty. The information content in survey forecasts

is examined again in Section 4.4.

4.3 The Impact of Data Revisions

Macroeconomic data go through substantial revisions after their initial release.12 Likewise,

macroeconometric analysis using latest available vintage data often results in different

conclusions from work that takes real-time issues into account (Orphanides 2001 and Or-

phanides and Van Norden 2002, for example). A large body of papers has also shown that

real-time data issues are particularly important for evaluating the forecasting power of

econometric models (Diebold and Rudebusch 1991, Faust et al. 2003, Amato and Swanson

2001, and Ghysels et al. 2014, among many others). However, previous studies focus on the

effects on point forecasts (i.e., the conditional mean); likewise, the effect of data revision on

the estimation of the conditional second moments has not been documented, to the best of

our knowledge. Nonetheless, data revisions should also have an important impact on the

measurement of uncertainty, since they directly affect the magnitude of forecast errors. Our

baseline measure is estimated with forecast errors computed using the first data release; in

this section, we also examine to what extent our macroeconomic uncertainty index differs,

if we use vintages available in one and five quarter(s) after the initial release as well as

12See Faust et al. (2005), Aruoba (2008) and Amir Ahmadi et al. (2015) for more details on the empiricalproperties of data revisions.

18

Figure 5: Macroeconomic Uncertainty Based on the Different Vintages

1970 1975 1980 1985 1990 1995 2000 2005 2010 20150

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8Initial Release2nd Vintage6th VintageFinal Vintage

Note: This figure compares the baseline macroeconomic uncertainty factor (the first release) with theuncertainty factor estimated with data from different vintages. The solid line is our baseline uncertaintyindex, the blue dashed line is the one where forecasting errors of each variable are computed using therevised data available in one quarter after the initial release, the green dash-dot line, is the 6th vintage,and the red line, the final vintage data.

the final 2015Q2 vintage to calculate the forecast errors. Hence, for the first time in the

literature, to the best of our knowledge, our paper examines the effects of data revisions

for the estimation of macroeconomic volatility.

Figure 5 shows the macro uncertainty index based on the first-released data (the baseline

measure), the data revised in one and five quarter(s) after the initial release, and the final

vintage data. The final vintage data differ from other vintages, as each data point has gone

through different numbers of revisions, that is, data from the further past have been revised

multiple times, while more recent data may have undergone only a couple of revisions.

The correlations among all four indexes are high, peaking substantially at the same

19

periods as the baseline index, i.e., the 1973–75, 1980, and 2008–09 recessions. Therefore,

we find that the estimated series of uncertainty using different data vintages largely coincide

under our framework. Nonetheless, we find quantitative differences across the four series.

Most notably, the relative size of the peaks changes depending on the data vintage used

to calculate forecasting errors. In other words, the jump in uncertainty during the 1980

recession is shifted upward when the second vintage is used, pushing the level of uncertainty

up to the highest level. In contrast, with the forecast errors based on the initial data release

and the sixth vintage, the uncertainty during the recessions in the pre-Great Moderation

periods, i.e., the 1973–75 and 1980 recessions, are higher than the level of uncertainty

during the last recession. However, the difference is most stark when the final vintage

data are used. In this case, the Great Recession is associated with the largest jump in

uncertainty since the beginning of the series, more in line with the dynamics of uncertainty

captured by the VIX and the Jurado et al. (2015) index.

In sum, our findings suggest that, while overall dynamics of both macroeconomic and id-

iosyncratic uncertainties remain robust, they exhibit differences, particularly in the relative

size of major peaks over time, depending on a particular data vintage chosen. For instance,

macroeconomic uncertainty based on the final vintage data will likely underestimate the

actual volatility faced by professional forecasters in the 1970s and 1980s in comparison with

the level during the Great Recession.

4.4 How Important Is the Information in the Survey?

It is important to remove predictable components from the economic indicators when es-

timating macroeconomic uncertainty so as not to attribute the predictable variation to

unpredictable shocks. As highlighted by Jurado et al. (2015), the most-commonly used

proxies of uncertainty —such as stock market volatility (i.e., VIX) and forecast dispersions

—do not account for this fact.

Several papers have documented that the predictive ability of survey forecasts is a

difficult benchmark for econometric forecasting models to beat, especially at short horizons

(see Ang et al. 2007; Faust and Wright 2013; Aiolfi et al. 2011). Consequently, survey

forecasts efficiently control for predictable variation in the economic indicators. To examine

20

the effectiveness of survey forecasts in removing predictable variations, here we re-estimate

our factor stochastic volatility model using forecasting errors from simple autoregressive

(AR) models:

yi,t = αi +

p∑j=1

βi,jyi,t−j + γ1y1i,t + γ2y

2i,t−1 + γ3y

3i,t−1 + εi,t, (6)

where p is the lag length chosen based on the Akaike information criterion with a rolling

fixed window of 60 observations. For the variables released at a monthly frequency, the

monthly values reported, i.e., the first-month value in quarter t (y1i,t, only if the release

date is before the survey deadline) and the third- and second-month values of quarter t− 1

(y3i,t−1 and y2i,t−1) are augmented to the AR model, to closely mimic information available

to professional forecasters at the time of the survey.13 It is worthwhile to note that the

real-time data available in each quarter t are used for the AR models to match our baseline

index more closely.

We then estimate stochastic volatility of a common factor using εi,t, the forecast errors

from the AR model above, for each of our four variables. The dotted line in Figure 6

plots the estimated macroeconomic volatility with the AR forecast errors. The AR-based

macroeconomic uncertainty index is considerably higher throughout the whole sample pe-

riod and more volatile than our baseline index based on survey forecasts. The result implies

that a significant share of the forecast errors from the AR models are indeed captured by

consensus forecasts, especially during recessions. Hence, simple statistical models such as

AR have not only larger forecasting errors but also more volatile ones, which are then

subsequently attributed to higher macroeconomic uncertainty.14 More importantly, this

analysis shows that using a consensus survey forecast is a parsimonious and effective way

of eliminating the predictable variations from economic variables, thereby providing a more

13The forecast errors from the above AR model are available at the Federal Reserve Bank of Philadelphiawebsite. In addition, Stark (2010) provides a detailed description regarding the model and real-timeavailable information sets.

14The Real-Time Data Research Center at the Federal Reserve Bank of Philadelphia provides statistics-related SPF forecast errors by comparing the forecast errors from the SPF with those from other simplebut widely used forecasting models: no-change forecast, iterative AR forecast, and horizon-specific ARforecast. The SPF forecast for t = 0, i.e., the nowcasts, of the four variables we use have consistentlylower RMSEs compared with other forecasting errors, and the differences are statistically significant. Thisfurther indicates that survey forecasts integrate wider sets of information than a simple statistical mode.

21

Figure 6: Macroeconomic Uncertainty: AR versus Surveys

1970 1975 1980 1985 1990 1995 2000 2005 2010 20150

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2BaselineAR−based

Note: This figure compares the baseline macroeconomic uncertainty index (estimated with survey forecasterrors) with the one estimated with errors obtained from the AR model as (6).

22

accurate measure of uncertainty faced by economic agents.

5 Innovations to Uncertainty and Macro Dynamics

In this section, we examine the dynamic relationship between our measure of macroe-

conomic uncertainty and a set of macroeconomic indicators using a standard recursively

identified VAR. Previous studies using proxies for macroeconomic uncertainty, such as the

VIX in Bloom (2009), tend to find a significantly negative, but short-lived impact of un-

certainty on economic activity, followed by an overshoot as economic activity rebounds. In

contrast, studies that estimate macroeconomic uncertainty, as Jurado et al. (2015), find

that these shocks have a much more persistent effect on economic activity, and no evidence

of a strong rebound and overshooting.

Our benchmark specification of a VAR comprises six variables, in the following order:

log(S&P index), uncertainty, log(private investment), log(non-durable consumption expen-

diture), log(durable consumption), log(GDP). The VAR is estimated in levels with four

lags. A natural choice of the ordering of variables is not clear, as our uncertainty measure

should react to real activity shocks within a quarter, while it is also possible that other

real variables respond to uncertainty during the same quarter. We follow previous litera-

ture and order our uncertainty series before the real activity indicators, but after the S&P

index.15

Figure 7 presents the estimated dynamic responses of the various economic activity

measures to a one standard deviation innovation to macroeconomic uncertainty. All of the

economic activity variables show a significant and persistent decline following the shock,

supporting the findings of long-lived negative effects of uncertainty as in Jurado et al. (2015)

and, to a lesser magnitude, Bachmann et al. (2013). In contrast to Bloom (2009), we find

no evidence of overshooting in economic activity as the uncertainty shock dissipates.

We further conduct a comparison of our impulse response functions with the ones ob-

tained with other estimates and proxies for macroeconomic uncertainty. Figure 8 plots

15See, for example, Bloom (2009), Jurado et al. (2015), Bachmann et al. (2013), Aastveit et al. (2013),and Caggiano et al. (2014). Our findings are robust to the changes of the variable ordering. Results fromdifferent versions of VARs are available upon request.

23

Figure 7: Responses to Innovations to Macroeconomic Uncertainty

Gross Domestic Product

Horizon

Per

cent

0 2 4 6 8 10 12 14 16 18 20−2

−1.5

−1

−0.5

0

0.5

1Private Investment

HorizonP

erce

nt0 2 4 6 8 10 12 14 16 18 20

−5

−4

−3

−2

−1

0

1

2

3

PCE Non−Durables

Horizon

Per

cent

0 2 4 6 8 10 12 14 16 18 20−1.5

−1

−0.5

0

0.5

1PCE Durables

Horizon

Per

cent

0 2 4 6 8 10 12 14 16 18 20−4

−3

−2

−1

0

1

2

3

Note: This figure shows the response of various macroeconomic variables to uncertainty shocks in a 6-variable VAR with the following variables and ordering: S&P index, macroeconomic uncertainty index, realprivate investment, real personal consumption expenditures on non-durables, real personal consumptionexpenditures on durables, real GDP. The VAR is estimated in levels with 4 lags. All variables, except theuncertainty index, are measured in logs. The shaded area represents two standard deviation bands usingKilian (1998)’s bootstrap-after-bootstrap method.

24

Table 2: Variance Decomposition for the Baseline VAR

Quarters RPI PCE-ND PCE-D GDP

1 4.64 6.42 9.99 6.092 10.25 11.46 19.95 16.074 15.15 21.87 23.85 21.708 19.67 27.16 24.47 25.3520 17.89 19.54 16.68 23.56

Note: This table shows forecast error variance decomposition for our baseline VAR with variables in the fol-lowing order: S&P index, uncertainty, real private investment (RPI), real personal consumption expenditureson non durables (PCE-ND), real personal consumption expenditures on durables (PCE-D), GDP. The VARis estimated in levels with 4 lags. All variables, except our uncertainty measure, enter the VAR in logs.

responses to innovations in our baseline uncertainty series, as well as the Jurado et al.

(2015) index (JLN), the average disagreement introduced in Section 4.2 and the VIX.

There is a clear divergence in results. Both innovations to our measures and the JLN mea-

sure of macroeconomic uncertainty result in large and persistent drops in economic activity.

On the other hand, innovations to proxies of macroeconomic uncertainty, the disagreement

and the VIX, result in very small and short-lived negative impacts on economic activity,

followed by strong rebounds.

To verify the quantitative importance of an uncertainty shock, Table 2 reports the

forecast error variance decomposition for the real economic activity variables included in

the VAR. The decomposition shows that an uncertainty shock can explain a maximum of

19.67 percent (real private investment) to 27.16 percent (non-durable consumption) of the

variance of various real economic activity measures within 5 years after the shock. These

numbers are in line with those reported by Bachmann et al. (2013) and Jurado et al. (2015).

6 Conclusion

This paper analyzes the evolution of economic uncertainty from 1968Q4 to 2015Q2 as per-

ceived by professional forecasters. Using a FSV framework proposed by Pitt and Shephard

(1999) permits us to jointly model and estimate time-varying macroeconomic and indicator-

specific uncertainties from the consensus forecast errors of different economic indicators.

Our baseline macroeconomic uncertainty index is persistent with all major spikes associ-

25

Figure 8: Responses with Alternative Measures of Uncertainty

Horizon

Per

cent

Gross Domestic Product

0 5 10 15 20−2

−1.5

−1

−0.5

0

0.5

1FSVJLNVIXDIS

HorizonP

erce

nt

Private Investment

0 5 10 15 20−4

−3

−2

−1

0

1

2FSVJLNVIXDIS

Horizon

Per

cent

PCE Non−Durables

0 5 10 15 20−1.5

−1

−0.5

0

0.5

1FSVJLNVIXDIS

Horizon

Per

cent

PCE Durables

0 5 10 15 20

−3

−2

−1

0

1

2FSVJLNVIXDIS

Note: This figure shows the response of various macroeconomic variables to uncertainty shocks in a 6-variable VAR using alternative uncertainty indexes: FSV is our baseline measure of macroeconomic un-certainty as perceived by professional forecasters; JLN is the measure proposed by Jurado et al. (2015);VIX is the stock market implied volatility used by Bloom (2009); and DIS is the average disagreement asdiscussed in Section 4.2. The shaded area represents one standard deviation bands using Kilian (1998)’sbootstrap-after-bootstrap method.

26

ated with deep economic recessions (the 1973–75, 1980, and 2007–09 recessions), consistent

with Jurado et al. (2015). However, other recessions (i.e., the 1990–91 and 2001 recessions)

are still notable in the dynamics of some of the idiosyncratic uncertainties but were not

picked up by the macroeconomic uncertainty series. Overall, we find that macroeconomic

uncertainty was higher in the 1968–85 period compared with the post-1985 period.

Using survey forecasts enables us to keep our framework simple and straightforward due

to their extensive information contents. Our baseline macroeconomic uncertainty index

shows dynamics similar to those of the Jurado et al. (2015) index, despite the substantial

differences in the number of underlying series. This highlights the informativeness of survey

forecasts and further their effectiveness in removing predictive variations from economic

indicators so as not to erroneously attribute them to unexpected shocks. A comparison

with an AR forecast-based index also confirms the efficiency of using survey forecasts, as

the AR-based uncertainty measure is considerably higher throughout the whole sample

period.

Furthermore, we document for the first time in the literature how data revisions affect

the estimated evolution of macroeconomic uncertainty. In particular, we show that the

relative size of major uncertainty peaks changes depending on a specific vintage chosen

to calculate forecast errors. We also conduct a VAR analysis to investigate the dynamic

relationship between uncertainty and real economic variables. A one-standard deviation

increase in our baseline uncertainty index results in a significant and persistent decrease

in various measures of economic activity such as investment, durable and non-durable

consumptions, in line with the findings in Bachmann et al. (2013) and Jurado et al. (2015).

However, this evidence is at odds with the short-lived negative impact followed by a strong

rebound, as suggested by Bloom (2009).

27

Appendix

A Bayesian Estimation Method

A.1 Prior Distributions and Starting Values

Our choice of prior distributions and their parameter values is very similar to that of

Pitt and Shephard (1999). The prior distribution for factor loadings is the Normal

distribution, i.e., λi ∼ N(λ0,Λ0) with λ0 = 1 and Λ0 = 25, as in Pitt and Shephard (1999).

The choice of relatively large Λ0 represents a fair degree of uncertainty around the factor

loadings. The initial value of the factor loadings is the OLS estimates of forecasting errors

on the first principal component as a proxy of a factor. Since the factor and loadings

are not completely identified in a factor model, we set the loading of the first variable

(i.e., GDP) to be equal to one, a commonly used identification strategy of a factor model.

Factor loadings of the second to the last variable are drawn from the posterior distribution

introduced below.

A diffuse Normal prior is used as the prior distribution for a factor conditional on

{hf,t}Tt=1, consistent with (3) (i.e., ft ∼ N(θ0,Θ0) where θ0 = 0 and Θ0 = hf,t). As

mentioned above, we use the first principal component for the initial iteration.

The prior for the variability of volatilities is the inverse Gamma, i.e., σ2f and σ2

i

∼ IG(v02, δ0

2). We set v0 = 1 and δ0 = 1, which makes the conditional prior distribution

flatter than that in Pitt and Shephard (1999) and more so than those in other recent studies

incorporating stochastic volatility (see, e.g., Primiceri 2005; Baumeister et al. 2013) to allow

for a large time variation for stochastic volatilities a priori. Compared with the above recent

studies, the total number of parameters to estimate is substantially smaller in our case.

Thus, we use a more diffuse prior and put a larger weight on data.

The prior of each time-varying volatility is the log-Normal. In particular, for the

initial period’s stochastic volatility, h20, we have log h0 ∼ N(µh0 , Vh0 ), where µh0 = 1 and

V h0 = 10 to allow a good chance for the data to determine the posterior distribution.

28

A.2 Posterior Distribution Simulation

The MCMC algorithm for the estimation of the joint posterior distribution of an FSV model

closely follows the one introduced in Pitt and Shephard (1999). We divide the parameters

in the model into four blocks: (a) the factor loadings (λ); (b) the time series of the factor

({ft}Tt=1); (c) the hyperparameters for volatilities (σf ; and σi for all i); and (d) the volatility

states ( {hf,t}Tt=1 and {hi,t}Tt=1 for all i). Due to the conditional independence across t and i

as well as of ft and ui,t’s, most steps in the Gibbs sampler, such as drawing factor loadings

({λ}), volatility states ({h}) and the variance of volatilities ({σ}) further break down to

drawing from a univariate process. Finally, the volatility states are drawn via Metropolis

methods within the overall Gibbs sampler. Denoting by zT the time series of a variable z

from t = 1 to T , the sampler algorithm is described below.

A.2.1 Factor loadings

Conditional on all other parameters, this step becomes a simple Bayesian regression of

forecasting errors on the factor with known heteroskedastic error structures. Moreover, be-

cause all correlations are by definition captured by the factor, this step further decomposes

into the n sub-steps of drawing each i-th loading separately from the following distribution,

given the history of variable i:

λi|εT , fT , σf , σi, hTf , hTi ∼ N(λ1,Λ1),

where Λ1 = (Λ−10 +∑T

t=1 f2t /(hi,t))

−1 and λ1 = Λ1(Λ−10 λ0 +

∑Tt=1 ft · εi,t/(hi,t)) with s ≥ 1

as the starting point of each forecasting error series εi.

A.2.2 Factor

Conditional independence also simplifies this step. Given all other parameter values, this

step again becomes a Bayesian regression of available forecasting errors on factor loadings

with known heteroskedasticity for each period t. That is,

ft|εT , λi, σf , σi, hTf , hTi ∼ N(θ1,Θ1),

29

where Θ1 = {h−1f,t +∑n

i=1 λ2i (hi,t)}−1, and θ1 = Θ1(

∑ni=1 λi · εi,t/(hf,t)).

A.2.3 Innovation variance of volatilities

Since we model each stochastic volatility to follow a unit-root process without a drift, the

conditional posterior distribution of σ can be simplified from the posterior inverse Gamma

distribution in Kim et al. (1998). Hence, σ2 is drawn from

σ|εT , fT , hTf , hTi ∼ IG(v12,δ12

),

where v1 = v0 + T, and δ1 = δ0 +∑T

t=1(hi,t − hi,t−1)2.

A.2.4 Volatility states

This step further decomposes to the n + 1 sub-steps of univariate stochastic volatility

draws, based on the Markovian property of stochastic volatility. It follows the algorithm

by Jacquier et al. (2002) as used in Cogley and Sargent (2005). For each volatility series of

an idiosyncratic error i or of the factor, the algorithm draws the exponential of volatility

(h2i,t) one by one for each t = 1, . . . , T , based on f(hi,t|hi,t−1, hi,t+1, yTi , λ, f

T , σ).

Before sampling the states, we first transform forecasting errors to be ε∗i,t = εi,t− λi,tft.

Such transformation is unnecessary for the factor. Then, we apply the Jacquier et al. (2002)

algorithm for each date, i.e.,

f(hi,t|(hi)T−t, y∗Ti , σ) = f(hi,t|hi,t−1, hi,t+1, y∗Ti , σ)

∝ f(y∗i,t|hi,t)f(hi,t|hi,t−1)f(hi,t+1|hi,t)

= (hi,t)−1.5 exp

(−y∗i,t2hi,t

)exp

(−(log hi,t − µt)2

2σ2c

),

where µt and σ2c are the conditional mean and variance of log hi,t, respectively. Under the

unit-root specification of this paper, they can be calculated as

µt =(log hi,t−1 + log hi,t+1)

2,

σ2c =

σ2i

2,

30

for t = 1, · · ·, T − 1. Thus, a trial value of log hi,t is drawn from the Normal distribution

with mean µt and variance σ2c . For the beginning and end periods of each series i, the

following conditional mean and variance are used instead:

t = 0 : σ2c =

σiVh0

σi+V h0, µ = σ2

c

(µh0V h0

+log hi,t+1

σ2i

),

t = T : σ2c = σ2

i , µ = log hi,T−1.

After obtaining a draw, the conditional likelihood f(y∗i,t|hi,t) is evaluated in order to obtain

the acceptance probability, completing a Metropolis step (see Cogley and Sargent 2005 for

a detailed description).

We can summarize the estimation procedure as following:

1. Assign initial values for λ, fT , σf , σi for all i, hTf , and hTi for all i.

2. Draw λ from p(λ|εT , fT , σf , σi, hTf , hTi ).

3. Draw fT from p(fT |εT , λ, σf , σi, hTf , hTi ).

4. Draw σf and σis from p(σ|εT , fT , hTf , hTi ).

5. Draw hTf , and hTi s from p(h|εT , fT , σf , σi).

6. Go to step 2.

We iterate over the Metropolis-within-Gibbs sampler a total of 80, 000 times, discarding

the first 30, 000 draws of parameters. Then we store every 10th draw in order to avoid po-

tential autocorrelation across draws, and finally obtain 5, 000 draws from the joint posterior

distribution.

References

Aastveit, K. A., G. J. Natvik, and S. Sola (2013): “Economic Uncertainty and theEffectiveness of Monetary Policy,” Norges Bank Working Paper Series, 17.

Aguilar, O. and M. West (2000): “Bayesian dynamic factor models and portfolioallocation,” Journal of Business & Economic Statistics, 18, 338–357.

Aiolfi, M., C. Capistran, and A. Timmermann (2011): “Forecast Combinations,”

31

in The Oxford Handbook of Economic Forecasting, ed. by M. Clements and D. Hendry,Oxford University Press, 355–88.

Alexopoulos, M. and J. Cohen (2009): “Uncertain Times, Uncertain Measures,”University of Toronto Department of Economics Working Paper, 352.

Amato, J. D. and N. R. Swanson (2001): “The real-time predictive content of moneyfor output,” Journal of Monetary Economics, 48, 3–24.

Amir Ahmadi, P., C. Matthes, and M.-C. Wang (2015): “Measurement Errors andMonetary Policy: Then and Now,” FRB Richmond Working Paper, 15–84.

Ang, A., G. Bekaert, and M. Wei (2007): “Do Macro Variables, Asset Markets, orSurveys Forecast Inflation Better?” Journal of Monetary Economics, 54, 1163–1212.

Arai, N. (2014): “Using forecast evaluation to improve the accuracy of the Greenbookforecast,” International Journal of Forecasting, 30, 12–19.

Arellano, C., Y. Bai, and P. J. Kehoe (2012): “Financial Frictions and Fluctuationsin Volatility,” Federal Reserve Bank of Minneapolis Research Department Staff Report,466.

Aruoba, S. B. (2008): “Data revisions are not well behaved,” Journal of Money, Creditand Banking, 40, 319–340.

Bachmann, R., S. Elstner, and E. Sims (2013): “Uncertainty and Economic Activity:Evidence from Business Survey Data,” American Economic Journal: Macroeconomics,5, 217–49.

Baker, S. R., N. Bloom, and S. J. Davis (2013): “Measuring Economic Policy Un-certainty,” Chicago Booth Research Paper, 13-02.

Baumeister, C., P. Liu, and H. Mumtaz (2013): “Changes in the Effects of MonetaryPolicy on Disaggregate Price Dynamics,” Journal of Economic Dynamics and Control,37, 543–560.

Bekaert, G., M. Hoerova, and M. L. Duca (2013): “Risk, uncertainty and monetarypolicy,” Journal of Monetary Economics, 60, 771–788.

Bloom, N. (2009): “The Impact of Uncertainty Shocks,” Econometrica, 77, 623–685.

Bloom, N., M. Floetotto, N. Jaimovich, I. Saporta-Eksten, and S. J. Terry(2012): “Really Uncertain Business Cycles,” NBER Working Paper, w18245.

Caggiano, G., E. Castelnuovo, and N. Groshenny (2014): “Uncertainty Shocksand Unemployment Dynamics: An Analysis of Post-WWII US Recessions,” Journal ofMonetary Economics, 67, 78–92.

32

Caldara, D., J. Fernandez-Villaverde, J. F. Rubio-Ramirez, and W. Yao(2012): “Computing DSGE models with recursive preferences and stochastic volatility,”Review of Economic Dynamics, 15, 188–206.

Carriero, A., T. E. Clark, and M. Marcellino (2015): “Common drifting volatilityin large Bayesian VARs,” Journal of Business & Economic Statistics, forthcoming.

Chib, S., F. Nardari, and N. Shephard (2006): “Analysis of high dimensional mul-tivariate stochastic volatility models,” Journal of Econometrics, 134, 341–371.

Chib, S., Y. Omori, and M. Asai (2009): “Multivariate stochastic volatility,” in Hand-book of Financial Time Series, Springer, 365–400.

Cogley, T. and T. J. Sargent (2005): “Drifts and Volatilities: Monetary Policies andOutcomes in the Post-WWII U.S.” Review of Economic Dynamics, 8, 262–302.

Collin-Dufresne, P. and R. S. Goldstein (2002): “Do bonds span the fixed incomemarkets? Theory and evidence for unspanned stochastic volatility,” The Journal ofFinance, 57, 1685–1730.

Del Negro, M. and C. Otrok (2008): “Dynamic factor models with time-varyingparameters: measuring changes in international business cycles,” FRB of New YorkStaff Report.

Diebold, F. X. and G. D. Rudebusch (1991): “Forecasting output with the compositeleading index: A real-time analysis,” Journal of the American Statistical Association, 86,603–610.

Faust, J., J. H. Rogers, and J. H. Wright (2003): “Exchange rate forecasting: theerrors we’ve really made,” Journal of International Economics, 60, 35–59.

——— (2005): “News and noise in G-7 GDP announcements,” Journal of Money, Creditand Banking, 37, 403–419.

Faust, J. and J. H. Wright (2013): “Forecasting Inflation,” in Handbook of EconomicForecasting, ed. by G. Elliott and A. Timmermann, Elsevier, vol. 2, Part A of Handbookof Economic Forecasting, 2–56.

Ghysels, E., C. Horan, and E. Moench (2014): “Forecasting through the rear-viewmirror: Data revisions and bond return predictability,” FRB of New York Staff Report.

Heston, S. L. (1993): “A closed-form solution for options with stochastic volatility withapplications to bond and currency options,” Review of Financial Studies, 6, 327–343.

Jacquier, E., N. G. Polson, and P. E. Rossi (2002): “Bayesian Analysis of StochasticVolatility Models,” Journal of Business & Economic Statistics, 20, 69–87.

Jo, S. (2014): “The Effects of Oil Price Uncertainty on Global Real Economic Activity,”Journal of Money, Credit and Banking, 46, 1113–1135.

33

Jorda, O. (2005): “Estimation and inference of impulse responses by local projections,”American Economic Review, 95, 161–182.

Jurado, K., S. C. Ludvigson, and S. Ng (2015): “Measuring Uncertainty,” AmericanEconomic Review, 105, 1177–1216.

Justiniano, A. and G. E. Primiceri (2008): “The Time-Varying Volatility of Macroe-conomic Fluctuations,” American Economic Review, 98, 604.

Kastner, G., S. Fruhwirth-Schnatter, and H. F. Lopes (2014): “Analysis ofExchange Rates via Multivariate Bayesian Factor Stochastic Volatility Models,” in TheContribution of Young Researchers to Bayesian Statistics, Springer, 181–185.

Kilian, L. (1998): “Small-Sample Confidence Intervals for Impulse Response Functions,”Review of Economics and Statistics, 80, 218–230.

Kim, C.-J. and C. R. Nelson (1999): “Has the US Economy Become More Stable? ABayesian Approach Based on a Markov-Switching Model of the Business Cycle,” Reviewof Economics and Statistics, 81, 608–616.

Kim, S., N. Shephard, and S. Chib (1998): “Stochastic Volatility: Likelihood Inferenceand Comparison with ARCH Models,” Review of Economic Studies, 65, 361–393.

Lahiri, K. and X. Sheng (2010): “Measuring Forecast Uncertainty By Disagreement:The Missing Ink,” Journal of Applied Econometrics, 25, 514–538.

Mankiw, N. G., R. Reis, and J. Wolfers (2004): “Disagreement About InflationExpectations,” in NBER Macroeconomics Annual 2003, Volume 18, The MIT Press,209–270.

McConnell, M. M. and G. Perez-Quiros (2000): “Output Fluctuations in the UnitedStates: What Has Changed since the Early 1980’s?” American Economic Review, 90,1464–1476.

Mumtaz, H. and P. Surico (2013): “Policy Uncertainty and Aggregate Fluctuations,”Working Paper, School of Economics and Finance, Queen Mary, University of London.

Mumtaz, H. and K. Theodoridis (2015): “Common and country specific economicuncertainty,” Working Paper, School of Economics and Finance, Queen Mary, Universityof London.

Mumtaz, H. and F. Zanetti (2013): “The impact of the volatility of monetary policyshocks,” Journal of Money, Credit and Banking, 45, 535–558.

Orphanides, A. (2001): “Monetary policy rules based on real-time data,” AmericanEconomic Review, 91, 964–985.

Orphanides, A. and S. Van Norden (2002): “The unreliability of output-gap estimatesin real time,” Review of Economics and Statistics, 84, 569–583.

34

Pitt, M. and N. Shephard (1999): “Time-Varying Covariances: A Factor StochasticVolatility Approach,” Bayesian Statistics, 6, 547–570.

Primiceri, G. E. (2005): “Time varying structural vector autoregressions and monetarypolicy,” Review of Economic Studies, 72, 821–852.

Rossi, B. and T. Sekhposyan (2015): “Macroeconomic uncertainty indices based onnowcast and forecast error distributions,” American Economic Review, 105, 650–655.

Scotti, C. (2013): “Surprise and Uncertainty Indexes: Real-Time Aggregation of Real-Activity Macro Surprises,” FRB International Finance Discussion Paper, 1093.

Sill, K. (2012): “Measuring Economic Uncertainty Using the Survey of Professional Fore-casters,” Federal Reserve Bank of Philadelphia Business Review, 4, 16–27.

Stark, T. (2010): “Realistic Evaluation of Real–Time Forecasts in the Survey of Profes-sional Forecasters,” Federal Reserve Bank of Philadelphia Special Report.

Zarnowitz, V. and P. Braun (1993): “Twenty-two years of the NBER-ASA quarterlyeconomic outlook surveys: Aspects and comparisons of forecasting performance,” inBusiness cycles, indicators and forecasting, University of Chicago Press, 11–94.

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